论文信息 - An explicit formula for Lanczos polynomials

An explicit formula for Lanczos polynomials

Abstract The Lanczos algorithm for tridiagonalizing a given matrix A generates a sequence of approximating matrices A p that can naturally be obtained as restrictions to subspaces. The eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of A . In this paper we produce explicit formulas for the characteristics polynomials of the A p in terms of the eigenvalues of A . These formulas can be used to explain heuristically why these approximations are often quite good. At present, we have no concrete analytic argument that explains the quality of the approximation. The main result draws on the formal relationship between the Lanczos algorithm and Pade approximations to the moment generating function of A . This result is one of the few analytic results available for the unsymmetric Lanczos algorithm.

George Cybenko | G. Cybenko

[1] B. Parlett. The Symmetric Eigenvalue Problem , 1981 .

[2] Zhishun A. Liu,et al. A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[3] Y. Saad. On the Rates of Convergence of the Lanczos and the Block-Lanczos Methods , 1980 .

[4] George Cybenko. Restrictions of Normal Operators, Padé Approximation and Autoregressive Time Series , 1984 .

[5] B. Parlett,et al. Lanczos versus subspace iteration for solution of eigenvalue problems , 1983 .

[6] Christopher C. Paige,et al. The computation of eigenvalues and eigenvectors of very large sparse matrices , 1971 .

[7] J. Cullum,et al. Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[8] S. Kaniel. Estimates for Some Computational Techniques - in Linear Algebra , 1966 .

[9] Alston S. Householder,et al. The Theory of Matrices in Numerical Analysis , 1964 .

[10] Beresford N. Parlett,et al. The Software Scene in the Extraction of Eigenvalues from Sparse Matrices , 1984 .

[11] W. Gragg,et al. The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis , 1972 .

[12] C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[13] J. Cullum,et al. Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory , 1984 .

[14] J. H. Wilkinson. The algebraic eigenvalue problem , 1966 .